Supersolid behavior of nonlinear

Albert Ferrando,1 Miguel-Angel´ Garc´ıa-March,2 and Mario Zacar´es2 1Departament d’Optica,` Universitat de Val`encia. Dr. Moliner, 50. E-46100 Burjassot (Val`encia), Spain.∗ 2Instituto Universitario de Matem´atica Pura y Aplicada (IUMPA), Universidad Polit´ecnica de Valencia. Camino de Vera s/n. E-46022 Valencia, Spain.∗ (Dated: April 18, 2019) We present a formal demonstration that light can simultaneously exhibit a superfluid behavior and spatial long-range order when propagating in a photonic crystal with self-focussing nonlinearity. In this way, light presents the distinguishing features of matter in a “supersolid” phase. We show that this supersolid phase provides the stability conditions for nonlinear Bloch waves and, at the same time, permits the existence of topological or defects for the envelope of these waves. We use a condensed matter analysis instead of a standard nonlinear approach and provide numerical evidence of these theoretical findings.

Matter can undergo a phase transition at ultracold fying ∂F/∂φ¯∗ = 0. For the case under consideration, the temperatures in which exhibits a superfluid behavior and, optical free would be given by (we use φ instead at the same time, present all the characteristics of a crys- of φ¯): talline solid. This new phase of matter is known as “supersolid” and it has been experimentally proven in helium-4 in recent years [1]. Apparently, light is unre- Z lated to these phases characteristic of condensed matter. 2 h ∗ 2 2 g 4i F = d x ∇tφ ∇tφ + V (x)|φ| − µ|φ| − |φ| . However, analogies between condensed matter and opti- 2 cal systems are increasingly appearing in the literature (1) [2]. Even the concept of liquid phase of light has been In this context the typical P (µ) curve for already suggested [3]. In this letter, we will take this a stationay solution appears as the conventional equation analogy a step further and we will demonstrate a formal of state of statistical physics P = −∂F/∂µ. equivalence between regular light structures propagating in a nonlinear photonic crystal and ultracold matter in a On the other hand, we are interested in the analysis “supersolid” phase. For the purpose of establishing this of stationary solutions whose amplitude is invariant un- equivalence, we will use the formalism of condensed mat- der finite translations of value a, where a is the period ter and [4] instead of using a standard of the potential. If φsol is a solution that satisfies the nonlinear optics approach. equation for stationary states and that simultaneously In Hamiltonian formalism the nonlinear Schr¨odinger verifies the condition |φsol(x + a)| = |φsol(x)| then the 2 equation for a periodic medium with Kerr nonlinear- total potential Vsol(x) ≡ V (x) − g|φsol(x)| occurring in ity can be obtained from the Hamiltonian density as such equation will be periodic with period given by a, ∗ 2 2 Vsol(x + a) = Vsol(x). Self-consistency implies that φsol i∂φ/∂z = ∂H/∂φ = −∇t + V (x) − g|φ| φ. In our must be equal to one of the Bloch functions φ (x) = case ∇t is the 2D transverse gradient operator and the sol iQ·x 2 2 fQ;β (x) = e vQ;β (x) forming the spectrum of the potential V (x) = − n (x) − n0 represents the periodic 0 0 modulation of the square of the with re- nonlinear operator generated by itself and it will be char- 2 acterized by a propagation constant µ = µQ;β . Using the spect to the reference value n0. Transverse and axial 0 coordinates are normalized. The energy of a propagat- nonlinear operator generated by φsol (including the total 2 arXiv:0808.0998v2 [nlin.PS] 10 Jul 2017 ing solution is given by E = R d2xH(φ). Now, instead of potential Vsol(x) ≡ V (x) − g|φsol(x)| ) we construct a using E we introduce the optical equivalent of the free nonlinear Wannier function basis by means of standard energy in statistical physics F = E − µP = R d2xF(φ) techniques. We assume that all functions are defined where µ is the propagation constant of a stationary so- within a periodic region Ω, called the basic domain, of lution and P is the system power (whose, in absence spatial dimensions Na × Na so that we represent any of losses, is a constant in the propagation). Thus P arbitrary field amplitude at a given axial position z as P µ plays the role of N, the particle number in statistical φ(x, z) = ˆi,α cˆi,α(z)Wα (x − xˆi).The nonlinear Bloch physics, and µ plays the role of the chemical potential. solution φsol(x) is represented by the site-independent sol iQ·x and z-independent coefficients c = η e ˆi where In terms of the free energy density the equation of motion √ ˆi µ ¯ ¯∗ 2 ¯ 2 ¯ is i∂φ/∂z = ∂F/∂φ = −∇t + V (x) − µ − g|φ| φ. It ηµ ≡ P /N. Note that Q is discretized because φ is de- is easy to check that the relation between the solutions of fined in the periodic domain Ω so that Qm = 2πm/(Na) ¯ −iµz 2 both equations is simply φ = φe . Thus an stationary where m ∈ Z and fulfills −N/2 ≤ mx, my ≤ N/2. We solution φ with propagation constant µ is equivalent to a introduce now the Wannier expansion into the expression z-independent solution φ¯ that is an extremum of F veri- for the optical free energy and integrate out the tranverse 2 coordinates x to obtain

2 X X F = −t(µ) c∗ c + c  + ˆi ˆi+ˆnν ˆi−nˆν ˆi ν=1 X U(µ) X +l(µ) |c |2 − |c |4 + (h.o.t.) , (2) ˆi 2 ˆi ˆi ˆi Figure 1: Effective potential for different configurations wheren ˆ1 andn ˆ2 are the “horizontal” and “vertical” lat- tice vectors, t(µ) ≡ −L = −L is the ef- 0ˆ,0+ˆˆ n1 0ˆ,0+ˆˆ n2 fective nearest neighbors coupling and l(µ) ≡ Lˆˆ and 00 F¯ ≡ sign(bm)F U(µ) ≡ 2T0ˆ0ˆ0ˆ0ˆ are the two on-site couplings. The second and fourth order couplings L (µ) and T (µ) are ob- ˆiˆj ˆiˆjkˆˆl Z tained as overlapping integrals of the nonlinear Wannier 2 ∗ ∗ F¯ = d x [|bm(µ)|∇Φ ∇Φ + isign(bm)(vm(µ) · ∇Φ)Φ basis associated to the stationary solution and, for this Ω reason, they depend on (µ, Q, β0) (for simplicity, we dis- +U(|Φ|2) + (h.o.t.) , (3) regard interband interaction terms and elliminate band indices): L (µ) ≡ R d2xW µ∗ −∇2 + V (x) − µ W µ where b (µ) ≡ cos(∆φ )t(µ)a2, v (µ) = v (1, 1) ˆiˆj Ω ˆi t ˆj m m m m 1 R 2 µ∗ µ∗ µ µ with vm(µ) ≡ −2 sin(∆φm)t(µ)a and Mm(µ) ≡ and Tˆˆˆˆ(µ) ≡ d xgW W W W . The expres- ijkl 2 Ω ˆi ˆj kˆ ˆl l(µ) − 4t(µ) cos(∆φ ). In this case (h.o.t.) in- sion (h.o.t.) in (2) stands for higher-order terms involving m cludes terms with higher-order derivatives and U = interactions at longer distances.  2 U(µ) 4 The main difference between (2) and previous ap- sign(bm) Mm(µ)|Φ| − 2 |Φ| . In principle, the use proaches is that the nonlinear Wannier basis provide dif- of F or F¯ is irrelevant as far as dynamics is concerned ferent coefficients than those obtained using linear Wan- since dynamics remains the same under the change F → nier functions or localized single potential solutions as in −F . The advantage in the use of F¯ is that it contains the tight binding aproximation. Since our aim is to anal- in all cases a positive definite kinetic energy term, as ize the stability behavior of the nonlinear Bloch solution required in quantum field theory to define a proper vac- characterized by µ the election of the nonlinear Wannier uum state [5]. By doing this, equivalences are straight- basis associated to it is a natural choice. With this pur- forward to achieve even with negative values of bm. By pose in mind let us write the z-dependent Wannier co- means of F¯ we can proceed to determine the nature of iQ·x iQ·x efficients as ci(z) = e ˆi Φi(z) = e ˆi (ηµ + ∆Φi(z)) the ground state of the system by analyzing the behav- to formalize the fact that we want to analyze the dy- ior of the so-called effective potential U(Φ) “`ala Lan- namics associated to the stationary solution described dau”. Landau theory is a mean-field approach used to sol iQ·xˆ by ci = ηµe i . In this way the dynamic information characterize phase transitions in condensed matter and is encoded in the envelope coefficients Φi(z). particle physics [4]. The qualitative form of the effective In this Letter we are interested in the regime where potential is represented in Fig. (1) for different signs of perturbations have a correlation length larger than the coefficients. One can easily recognize in this figure that potential period —ζ  a. Physical phenomena in this there are only two configurations for which a spatially regime present a collective spatial character that will re- uniform envelope solution Φ(x) = Φ0 is allowed (cases flected in the properties of the discrete envelope function (b) and (d)). They correspond to extrema of the opti- ∗ ∗ Φi(z). Spatial collective effects will be represented by cal free energy density ∂F/∂Φ = ∂U/∂Φ = 0 given by values of Φi that will fluctuate smoothly in space. It the condition |Φ0| = Mm(µ)/U(µ). The ground state of is then natural to take the continuous limit of the dis- the system is the state that minimizes the free energy, crete optical free energy by considering the limit a/ζ → 0 thus, only the two cases in (d) can provide an spatially and by introducing the continuous spatial envelope func- homogeneous ground state. The former analysis is iden- tion Φ(xt, z) defined as Φ(xˆi, z) = Φi(z). We substitute tical to that performed in to iQ·x ci = e ˆi Φi into (2) and then take the continuous limit establish the nature of phase transitions in the mean-field using standard techniques. In order to simplify the re- regime, here µ playing the role of temperature. In this sult, we consider only “diagonal” nonlinear Bloch waves way, the signs of the bm(µ), Mm(µ) and U(µ) coefficients with Q1 = Q2 = 2πm/(Na), m ∈ Z. The phase dif- determine the nature of the ground state and, therefore, ference between two neighboring sites of the nonlinear in which“phase” light is. It is important to strees that Bloch wave is then ∆φm = 2πm/N. In order to relate the µ dependence in these “Landau coefficients” is the re- the optical effects driven by F to condensed matter and sult of using nonlinear Wannier functions. The ground particle physics phenomena we need to re-define the op- state in Fig.1(d) is degenerate since all solutions of the iφ tical free energy in some cases. For this reason, we in- type Φφ = |Φ0|e are minima of the effective poten- troduce the sign-changed optical free energy defined as tial and thus they all have the same free energy. The 3 optical free energy is invariant under a U(1) phase trans- iα formation Φ → e Φ. The ground state Φφ, however, is not since this phase transformation maps it into a differ- ent degenerate solution Φφ+α with the same free energy. This mechanism is well-known in condensed matter and particle physics and it is known as spontaneous symme- try breaking (SSB). Superfluidity, superconductivity, the Higgs mechanism or the chiral phase transition in quan- tum chromodynamics are physical phenomena related to Figure 2: Landau coefficients (bm, Mm, U) in terms of the the same mechanism. The SSB mechanism have distinc- propagation constant µ for unstaggered (dashed line) and tive properties: (i) appearence of a non-zero order pa- staggered (solid line) nonlinear Bloch solutions. rameter in the broken phase, (ii) existence of topological solitons or defects, (iii) presence of masless or long-range (a) (b) excitations (Goldstone bosons). In this language, our op- tical system in the Fig.1(d) configuration is in a broken phase, a phase in which U(1) symmetry has been spon- taneously broken. We expect then to find the optical counterparts of the aforementioned properties. In this Letter we will pay attention to properties (i) and (ii), leaving the analysis of (iii) for a further publication. We can now establish a link between the stability of Figure 3: Stability/unstability patterns of phase under prop- nonlinear Bloch waves and the SSB mechanism. A non- agation:(a) for a perturbed unstaggered solution phase disor- linear Bloch wave is characterized by an homogeneous ders quickly, (b) for a staggered solution spatial order remains. envelope. According to our previous analysis, only in the configurations in Fig.1(d) this envelope function cor- responds to the ground state of the optical free energy. nonlinear photonic crystal with C symmetry formed by On the other hand, in this configuration the system is 4 a nonlinear material with refractive index n with em- in the U(1) broken phase. Since the ground state is a 1 bedded circular inclusions of a linear material with index stable solution, i.e., fluctuations around it cannot trans- n < n , the index contrast being given by the potential form this solution into a different one by any dynamical 2 1 V = − n2 − n2 > 0. This type of nonlinear photonic mechanism, we infer that the stability condition of a non- 0 2 1 crystal can be achieved in standard or in chalcogenide linear Bloch wave is achieved in the broken phase, i.e., photonic crystal fibers [6] or in -writing waveguides when (b > 0, M < 0, U < 0) or (b < 0, M > 0, m m m m [7]. The nonlinearity is self-focussing and of the Kerr type U > 0). The properties of a stable nonlinear Bloch wave V = −g|φ|2 (g > 0). First of all, we have evaluated the are then identical to that of a superfluid as far as its enve- NL dependence of the “Landau coefficients” (b , M , U) on lope is concerned. However, the solution simultaneously m m µ . We have proceeded as follows: (i) we numerically presents spatial long-range order. Using quantum field evaluate a nonlinear Bloch solution φ at a given µ, (ii) theory notation, if |0i stands for the ground state given sol we determine the nonlinear operator associated to φ by the nonlinear Bloch wave in the mean-field regime sol (i.e., including the nonlinear potential V = −g|φ |2) at a given µ: (i) T |0i = eiQ·a |0i where T is the lat- NL sol a a which numerically will be a matrix, (iii) we find the spec- tice translation operator and (ii) h0| Φˆ |0i = |Φ |eiφ 6= 0. 0 trum of this matrix formed by Bloch eigemodes since the Physically speaking, the “ lattice” filling the pe- riodic medium completely and described by the nonlin- ear Bloch wave presents perfect spatial long-range order (a) (b) (c) as in a cristallyne solid and, at the same time, it pos- sesses a non-vanishing order parameter indicating it is in a superfluid phase. The dynamics of low energy fluctua- tions around such a solution is determined by Eq.(3) and has to show identical features than a superfluid. In this way, we demonstrate that, under the specified conditions, light fulfills the definition of a supersolid, that is, it is a Figure 4: Supersolid light topological defect for µ = −0.1, spatially ordered system (like in a solid or crystal) with V0 = 2, and charge +1: (a) simultaneous representation of superfluid properties. the amplitudes of the envelope Φ and full φ functions for this We have performed a number of numerical experiments solution, (b) representation of its phase and (c) P (µ) diagram to check both qualitative and quantitative the validity of calculated using the envelope equation (solid line) and the full our condensed matter approach. We have modelled a equation (triangles). 4 total potential, including the linear and nonlinear part, is Fig.4(b)— which corresponds to the ground state in the periodic, (iv) we construct the nonlinear Wannier basis broken superfluid phase |Φ0| 6= 0. From an optical point out of these Bloch modes using the standard technique of view, one could consider this object as a type of dark [8] and (v) we evaluate the “Landau coefficients” using soliton, however, the striking property of this dark soli- their definitions as overlapping integrals of Wannier func- ton is that exists in a self-focusing medium when they tions. A given nonlinear Bloch wave characterized by µ are usually associated to defocusing media. In Fig.4(a) then univocally provides a specific value for (bm, Mm, we can appreciate the excellent fit of the envelope func- U). In Fig. 2 we represent the functional form of these tion to the solution of the full equation. This excellent coefficients in terms of µ for two solutions with differ- quantitative agreement is even more clearly appreciated ent pseudo-momentum Q: a solution with Q = 0 with in the calculation of the P (µ) curve for a family of topo- a phase difference between neighboring sites ∆φm = 0 logical vortices using both approaches shown in Fig.4(c). (unstaggered) and a solution with Q = (π/a, π/a) with In summary, Figs.4(a) and (b) show the simultaneous ∆φm = π (staggered). By analyzing the signs of their presence of superfluid behavior (nontrivial amplitude and Landau coefficients we immediately recognize these two phase of the topological defect) and spatial long-range or- configurations correspond to the cases (b) (unstaggered) der (perfect staggered order of the background in all the and (d) (staggered) in Fig.1. The main difference be- domain). Besides, numerical stability analysis indicates tween these two configurations arise from the different that these solutions are stable under propagation during sign of bm, which has its origin in their different phase long distances. These features represent a clear numerical structure since bm ∼ t(µ) cos(∆φm). In the case ana- evidence of the supersolid behavior of light in nonlinear lyzed here t(µ) > 0 for all µ in both configurations, so photonic crystals predicted by theory. In this sense, it that the different form of the effective potential in both has been shown that light can be treated analogously as cases is a pure effect of the underlying phase of the non- matter not only in their qualitative aspects but by using linear Bloch wave. According to our condensed matter a condensed matter formalism. We call this approach to analysis, the unstaggered solution can exist as an homo- nonlinear optics photonic condensed matter. geneous solution but it corresponds to a metastable state that eventually will decay into a different state. It cannot This work was partially supported by the Gov- be stable. On the contrary, the staggered solution is the ernment of Spain (contracts FIS2005-01189 and ground state of a light supersolid and, consequently, it is TIN2006-12890) and Generalitat Valenciana (contract necessary stable. We have performed a numerical stabil- APOSTD/2007/052). ity analysis of these two configurations that confirm these predictions. A small initial perturbation of the unstag- gered solution gives rise, after a short propagation, to the breaking of spatial long-range order crearly reflected in the progressive disordering of the phase of the solution — ∗ Interdisciplinary Modeling Group, InterTech see Fig.3(a). The staggered solution is, however, immune (http://www.intertech.upv.es) to perturbations and preserve spatial long-range order in [1] E. Kim and M. H. W. Chan, Nature 427, 225 (2004). phase and amplitude —see Fig.3(b). Similar examples [2] B. Freedman et al, Nature 440, 1166 (2006); T. Schwartz et al, Nature 446, 52 (2007); T. Pertsch et al, Phys. Rev. can be found in the literature that can be explained by Lett. 83, 4752 (1999); R. Morandotti et al, Phys. Rev. the same mechanism, e.g., stability of large truncated Lett. 83, 4756 (1999). H. Martin et al, Phys. Rev. Lett. nonlinear Bloch waves [9]. As mentioned before, one of 92, 123902 (2004). the most paradigmatic properties of a superfluid is the ex- [3] H. Michinel et al, Phys. Rev. E 65, 066604 (2002). istence of topological solitons or defects. They are holes [4] P. M. Chaikin and T. C. Lubensky, Principles of Con- in the superfluid around which the superfluid flows with a densed Matter Physics (Cambridge University Press, quantized circulation. Therefore, it is expected that the 2000). [5] S. Coleman, Aspects of symmetry: selected Erice Lectures envelope function Φ supports the existence of the optical (Cambridge University Press, 1985). counterparts of these objects in the broken phase. We [6] A. Ferrando et al, Opt. Express 11, 452 (2003); A. Fer- have proven indeed the existence of these type of solu- rando et al, Opt. Express 12, 817 (2004); L. Brilland et al, tions by numerically solving the same nonlinear photonic Optics Express 14, 1280 (2006). using: (i) the effective equation for Φ [7] D. Bl¨omer et al, Optics Express 14, 2151 (2006); A. Sza- meit et al, Optics Express 14, 6055 (2006). associated to F¯ —Eq.(3)— using the values of (bm, Mm, U) given in Fig.(2) and (ii) the full equation for the orig- [8] N. W. Ashcroft and N. D. Mermin, Solid state physics (Saunders College Publishing, 1976). inal field φ including the periodic potential V (x). It is [9] M. Petrovi´c et al, Physical Review E 68, 055601(R) remarkable that both approaches are in excellent agree- (2003); T. Alexander and Y. Kivshar, B: ment with each other both qualitatively and quantita- and Optics 82, 203 (2006). D. Tr¨ager et al, R. Fis- tively. As correctly predicted by theory, the topological cher, D. Neshev, A. Sukhorukov, C. Denz, W. Kr´olikowski, vortex only exists on top of the staggered solution —see and Y. Kivshar, Optics Express 14, 1913 (2006).